Phenomenology of Majorana zero modes in full-shell hybrid nanowires

Full-shell nanowires have been proposed as an alternative nanowire design in the search of topological superconductivity and Majorana zero modes (MZMs). They are hybrid nanostructures consisting of a semiconductor core fully covered by a thin superconductor shell and subject to a magnetic flux. Compared to their partial-shell counterparts, full-shell nanowires present some advantages that could help to clearly identify the elusive Majorana quasiparticles, such as the operation at smaller magnetic fields and low or zero semiconductor $g$ factor, and the expected appearance of MZMs at well-controlled regions of parameter space. In this paper we critically examine this proposal, finding a very rich spectral phenomenology that combines the Little-Parks modulation of the parent-gap superconductor with flux, the presence of flux-dispersing Caroli–de Gennes–Matricon (CdGM) analog subgap states, and the emergence of MZMs across finite flux intervals that depend on the transverse wavefunction profile of the charge density in the core section. Through microscopic simulations and analytical derivations, we study different regimes for the semiconductor core, ranging from the hollow-core approximation, to the tubular-core nanowire appropriate for a semiconductor tube with an insulating core, to the solid-core nanowire with the characteristic dome-shaped radial profile for the electrostatic potential inside the semiconductor. We compute the phase diagrams for the different models in cylindrical nanowires and find that MZMs typically coexist with CdGM analogs at zero energy, rendering them gapless. However, we also find topologically protected parameter regions or islands with gapped MZMs. In this sense, the most promising candidate to obtain topologically protected MZMs in a full-shell geometry is the nanowire with a tubular-shaped core. Moving beyond pristine nanowires, we study the effect of mode mixing perturbations. On the one hand, mode mixing can gap CdGM analogs and open minigaps around existing MZMs. On the other hand and rather strikingly, mode mixing can act like a topological $p$-wave pairing between particle-hole Bogoliubov partners, and is therefore able to create new topologically protected MZMs in regions of the phase diagram that were originally trivial. As a result, the phase diagram is utterly transformed and exhibits protected MZMs in around half of the parameter space.

Figure 1

Full-shell hybrid nanwowire. (a) Sketch of a full-shell nanowire in a cylindrical approximation. A semiconductor core (yellow) of radius $R$ is surrounded by a thin superconductor shell (blue) of thickness $d$. For modellistic purposes, the core can have a tubular shape of thickness $W$. In an applied axial magnetic field $B$ the hybrid wire is threaded by a nonquantized magnetic flux $\mathrm{\Phi }=\pi {(R+d/2)}^{2}B$. (b) Sketch of the radial dependence of the pairing amplitude $\mathrm{\Delta }\left(r\right)$ (blue), electrostatic potential energy $U\left(r\right)$ (green), and typical subgap wavefunction density ${\left|\mathrm{\Psi }\left(r\right)\right|}^2$ (yellow for the lowest radial mode $m_r=0$ and pink for the first radial mode $m_r=1$). The conduction-band bottom inside the semiconductor exhibits a dome-like radial profile with maximum value at the center $U_{\max }$, and minimum value at the superconductor-semiconductor interface $U_{\min }$. The electrostatic potential of the metallic shell is $|U_{\mathrm{shell}}|\gg |U_{\min }|$. (c) Schematics of a full-shell nanowire-based normal-superconductor junction, characterized by a tunnel potential barrier in the uncovered semiconductor region between the normal metal (N) and the full-shell wire (S). In red, Majorana bound state wavefunction at the end of the semi-infinite full-shell hybrid nanowire along the longitudinal direction. ${\xi }_{\mathrm{M}}$ is the Majorana localization length.

Figure 2

Hollow-core model. (a) (Left) Local density of states (LDOS) at the end of a semi-infinite $R=70\mathrm{nm}$ hollow-core nanowire (in arbitrary units) as a function of energy $\omega$ and applied normalized flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. The right half of the $n=0$ and the full $n=1$ LP lobes are displayed. The wire is in the nondestructive LP regime. (Right) LDOS of the $n=1$ lobe showing only the contribution of the $m_J=0$ subbands. Here, the SO coupling $\alpha =0$ and the Fermi energy $\tilde{\mu }=0.75\mathrm{meV}$. [(b)–(d)] Same as (a) but for different values of $\alpha$ marked by colored dots in (g). All the CdGM analogs coalesce at degeneracy points, located at the center of the LP lobes in the hollow-core approximation. (e) Band structure in the normal state (with decay rate into the shell ${\mathrm{\Gamma }}_{\mathrm{S}}\to 0$) as a function of the longitudinal wave vector $k_z$, corresponding to the case (d) at $\mathrm{\Phi }=0.51{\mathrm{\Phi }}_0$. The number of occupied subbands depends on the Fermi energy $\tilde{\mu }$. The chemical potential $\mu$ is marked with a black-dashed line. Different colors correspond to different values of the generalized angular momentum $m_J$ in the $n=1$ LP lobe. The $m_J=0$ sector gives rise to MZMs when proximitized. (f) $m_J=0$ LDOS corresponding to the case (d) but fixing the fluxoid number to $n=1$ for all the magnetic-flux range displayed. The fluxes at which the $n=1$ metastable wire enters and exits the topological phase, ${\mathrm{\Phi }}_{\mathrm{TT}}^{\left(i\right)}/{\mathrm{\Phi }}_0$ with $i=1,2$ [see Eq. (2)], are marked with arrows. (g) Topological phase diagram as a function of $\alpha$ and $\tilde{\mu }$ (for $R=70\mathrm{nm}$). The color scale represents the flux interval $L_{\mathrm{\Phi }}$ of the left Majorana ZEP within the first LP lobe, see Eq. (4). Red curves enclose regions of parameters, or islands, with topologically protected Majorana ZEPs (i.e., MZMs with a minigap for some flux in the $n=1$ lobe). (h) Same as (g) but as a function of ${\mathrm{\Gamma }}_{\mathrm{S}}$ and $\tilde{\mu }$ (for $\alpha =85\mathrm{meV}\mathrm{nm}$). (i) Same as (g) but as a function of $\alpha$ and $R$ (with $\tilde{\mu }=0.75\mathrm{meV}$). For $R$ smaller than the vertical dashed line, the wire is in the destructive LP regime. Other parameters: $d=0, {\mathrm{\Delta }}_0=0.23\mathrm{meV}, {\xi }_{\mathrm{d}}=70\mathrm{nm}, m^{*}=0.023m_e, {\mathrm{\Gamma }}_{\mathrm{S}}={\mathrm{\Delta }}_0, g=0$ and $a_0=5\mathrm{nm}$.

Figure 3

Tubular-core model. (a) Sketch of a tubular-core nanowire with shell thickness $d$, semiconductor-core radius $R$, and thickness $W$. [(b)–(f)] Local density of states (LDOS) at the end of a semi-infinite $R=70$ nm, $d\to 0$ tubular-core nanowire (in arbitrary units) as a function of energy $\omega$ and applied normalized flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$, displaying half of the $n=0$, and the full $n=1,2$ LP lobes. Panel (b) corresponds to the hollow-core approximation ($W\to 0$) and from (c) to (f) the thickness of the semiconductor tube increases in steps of $10\mathrm{nm}$. Parameters: $\alpha$ and $\mu$ are chosen to remain approximately at the same spot within the wedged-shaped topological region of the different-$W$ topological phase diagrams, see upper row of Fig. 4. ${\mathrm{\Gamma }}_{\mathrm{S}}$ is chosen so that the degeneracy points in the $n=0$ LP lobe remain fixed around $\omega =\pm 0.1\mathrm{meV}$, see lower row of Fig. 4. For example, in panel (f) $\alpha =20\mathrm{meV}\mathrm{nm}, \mu =7.1\mathrm{meV}, {\mathrm{\Gamma }}_{\mathrm{S}}=48{\mathrm{\Delta }}_0$, and $E_{\mathrm{g}}=52\mu \mathrm{eV}$ at $\mathrm{\Phi }=0.9{\mathrm{\Phi }}_0$. Other parameters as in Fig. 2.

Figure 4

Topological phase diagram for the tubular-core model. [(a)–(e)] Topological phase diagrams as a function of SO coupling $\alpha$ and chemical potential $\mu$ for tubular-core nanowires with $R=70\mathrm{nm}, d\to 0$ and increasing values of tube thickness $W$. In each panel ${\mathrm{\Gamma }}_{\mathrm{S}}$ corresponds to the white dot in the panel below. The color bar represents the Majorana localization length ${\xi }_{\mathrm{M}}$ at $\mathrm{\Phi }=0.51{\mathrm{\Phi }}_0$. The white dot in each panel corresponds to the parameters used in Figs. 3–3, respectively. The red islands contain parameters with topologically protected MZMs. [(f)–(j)] Same as (a)–(e) but as a function of decay rate ${\mathrm{\Gamma }}_{\mathrm{S}}$ and chemical potential $\mu$, with $\alpha$ fixed to the white dot in the panel above. The solid orange curves in (c) and (h) (that contours the topological regions) have been calculated using the modified hollow-core approximation with $R_{\mathrm{av}}=59.5\mathrm{nm}$. For these curves, $\tilde{\mu }\in [-2.7,3.3]\mathrm{meV}$ and ${\mathrm{\Gamma }}_{\mathrm{S}}^{\mathrm{av}}/{\mathrm{\Delta }}_0\in [0,2.1]$. Rest of parameters as in Fig. 2.

Figure 5

Modified hollow-core model. (a) Sketch of a tubular-core nanowire corresponding to Fig. 3, with $R=70\mathrm{nm}, W=20\mathrm{nm}, d=0, \alpha =50\mathrm{meV}\mathrm{nm}, \mu =18.1\mathrm{meV}$, and ${\mathrm{\Gamma }}_{\mathrm{S}}=12{\mathrm{\Delta }}_0$. The LP radius is $R_{\mathrm{LP}}=R+d/2$. Inside the core, simulation of the radial dependence of the electron density $n_e$, whose maximum value occurs at the average radius $R_{\mathrm{av}}$. (b) Wavefunction modulus of the populated $m_J$ subbands at $k_z=0$ in the normal state (${\mathrm{\Gamma }}_{\mathrm{N}}\to 0$) as a function of radial coordinate $r$. The wavefunction average radius is $R_{\mathrm{av}}=\langle r\rangle$. (c) Fitting of Fig. 3 using the modified hollow-core model with $R_{\mathrm{av}}=59.5\mathrm{nm}$ extracted from (b), $\tilde{\mu }=0.5\mathrm{meV}$ and ${\mathrm{\Gamma }}_{\mathrm{S}}^{\mathrm{av}}=1.1{\mathrm{\Delta }}_0$. The topological minigap is $E_{\mathrm{g}}=30\mu \mathrm{eV}$ at $\mathrm{\Phi }=0.71{\mathrm{\Phi }}_0$. The topological transition happens at $\mathrm{\Phi }=0.84{\mathrm{\Phi }}_0$. The $m_J$ quantum numbers of the different GdGM analogs are shown in the $n=1,2$ lobes. $L_{\mathrm{\Phi }}$ signals the MZM flux interval given in Eq. (6). (d) Same as (c) but for a finite shell thickness $d=10\mathrm{nm}$ and $g$ factor $g=10$. Now $E_{\mathrm{g}}=40\mu \mathrm{eV}$ at $\mathrm{\Phi }=0.84{\mathrm{\Phi }}_0$. The topological transition happens at $\mathrm{\Phi }=1.1{\mathrm{\Phi }}_0$. (e) Differential conductance $dI/dV$ corresponding to (d) (in units of the conductance quantum $G_0$) versus normalized flux for a device like the one represented in Fig. 1 with tunnel potential barrier of height $10\mathrm{meV}$ and length $50\mathrm{nm}$. Other parameters as in Fig. 2.

Figure 6

Solid-core model, lowest radial mode. (a) Sketch of the solid-core nanowire with $R=70\mathrm{nm}$ and $d=10\mathrm{nm}$. (b) Electrostatic potential energy displaying a dome-shaped radial profile inside the semiconductor core, with $U_{\max }=0\mathrm{meV}, U_{\min }=-30\mathrm{meV}$ and $\mu =-11\mathrm{meV}$. The chemical potential is such that only the lowest radial subband $m_r=0$ is populated. (c) Band structure in the normal state (${\mathrm{\Gamma }}_{\mathrm{S}}\to 0$) as a function of the longitudinal wave vector $k_z$. The number of occupied subbands depends on $\mu$. Different colors signal different values of the quantum number $m_J$ in the $n=1$ LP lobe. The average SO coupling is $\langle \alpha \rangle =20\mathrm{meV}\mathrm{nm}$ [corresponding to a prefactor ${\alpha }_0=46.7{\mathrm{nm}}^2$ in Eq. (A7)]. (d) LDOS at the end of a semi-infinite solid-core nanowire (in arbitrary units) as a function of energy $\omega$ and applied normalized flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$. There is a topological minigap $E_{\mathrm{g}}=15\mu \mathrm{eV}$ at $\mathrm{\Phi }=0.65{\mathrm{\Phi }}_0$ (marked with a dashed-white line). The parameters of the wire correspond to the pink dot in the topological phase diagram of Fig. 8. (e) Radial dependence of the previous LDOS at that flux. (f) Wavefunction modulus of the populated subbands at $k_z=0$ in the normal state (${\mathrm{\Gamma }}_{\mathrm{N}}\to 0$) as a function of radial coordinate $r$. (g) LDOS in the $n=1$ lobe coming only from the $m_J=0$ subband. For panels (c), (f), $\mathrm{\Phi }=0.65{\mathrm{\Phi }}_0$. Parameters: ${\mathrm{\Gamma }}_{\mathrm{S}}=40{\mathrm{\Delta }}_0, g=10$. Other parameters as in Fig. 2.

Figure 7

Solid-core model, first radial mode. Same as Fig. 6 but for a positive chemical potential, $\mu =2\mathrm{meV}$, such that not only the lowest radial subband $m_r=0$ is populated, but a few $m_J$ subbands have $m_r=1$. There is no topological minigap in this case for any flux. The parameters of the wire correspond to the white dot in the topological phase diagram of Fig. 8. For panels (c), (e), and (f), $\mathrm{\Phi }=1.49{\mathrm{\Phi }}_0$.

Figure 8

Topological phase diagram for the solid-core model. Map of ZEPs (a) and topological phase diagram (b) as a function of radially averaged SO coupling $\langle \alpha \rangle$ and potential profile minimum $U_{\min }$, with $U_{\max }=0$ and $\mu =0\mathrm{meV}$. The color scale in (a) represents the Majorana localization length ${\xi }_{\mathrm{M}}$ at $\mathrm{\Phi }=0.51{\mathrm{\Phi }}_0$. The color bar in (b) represents the number of MZMs $N_{\mathrm{M}}$, related to the topological invariant by $\mathcal{Q}={(-1)}^{N_{\mathrm{M}}}$ (red is nontrivial). [(c),(d)] Same as (a) and (b) but as a function of a rigid shift of the chemical potential $\mu$, fixing $U_{\min }=-30\mathrm{meV}$. Colored dots mark parameter values for Figs. 6 (orange) and 7 (white). Other parameters: $R=70\mathrm{nm}, d=10\mathrm{nm}, {\mathrm{\Delta }}_0=0.23\mathrm{meV}, {\xi }_{\mathrm{d}}=70\mathrm{nm}, {\mathrm{\Gamma }}_{\mathrm{S}}=40{\mathrm{\Delta }}_0, g=10$, and $a_0=5\mathrm{nm}$.

Figure 9

Hexagonal cross-section tubular-core model. (a) Topological phase diagram as a function of SO coupling $\alpha$ and Fermi energy $\tilde{\mu }$. The color bar represents the number of MZMs $N_{\mathrm{M}}$. [(b)–(e)] LDOS at the end of a semi-infinite nanowire as a function of $\omega$ and $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ in the $n=1$ LP lobe for the four colored points in the phase diagram of (a) (coded by the color of the cross section profile). (b) is indistinguishable from a cylindrical case with the same parameters. (c) exhibits two noninteracting MZMs, so their ZEPs add up, while in (e) they hybridize and their ZEPs split. (d) has a single MZM arising from a $m_J=\pm 3$ mode mixing. Parameters: (b) $\tilde{\mu }=0.35\mathrm{meV}, \alpha =35\mathrm{meV}\mathrm{nm}, E_{\mathrm{g}}=10\mu \mathrm{eV}$, (c) $\tilde{\mu }=0.5\mathrm{meV}, \alpha =60\mathrm{meV}\mathrm{nm}, E_{\mathrm{g}}=20\mu \mathrm{eV}$, (d) $\tilde{\mu }=1.5\mathrm{meV}, \alpha =35\mathrm{meV}\mathrm{nm}, E_{\mathrm{g}}=5\mu \mathrm{eV}$, (e) $\tilde{\mu }=1\mathrm{meV}, \alpha =145\mathrm{meV}\mathrm{nm}$. Other parameters as in Fig. 5.

Figure 10

Mode mixing in the modified hollow-core model. (First row) (a) Equivalent LDOS to Fig. 5 but for a SO coupling $\alpha =100$ meV nm, so that the left ZEP of the $n=1$ LP lobe is covered by several CdGM analogs. [(b)–(d)] Blow ups around zero energy $\omega$ of the $n=0$ (b), $n=1$ (c), and $n=2$ (d) LP lobes. The $n=1$ Majorana ZEP is visible in (c). There is no topological minigap for any flux. The quantum numbers $m_J$ are highlighted in (b) and (c). (Second row) Same as first row but for a modified hollow-core nanowire subject to a mode mixing of $\sigma /R_0=0.2$, using a smooth disorder model in Eq. (A27). The disorder-induced wavefunction distortion generates ZEPs at the edges of the $n=0$ lobe, see (f), and a long ZEP at the left side of the $n=2$ LP lobe, see (h). There is a small topological minigap in the three lobes. (Third row) Same as second row but with a nonsmooth disorder model representing atomic-sized defects. The ZEPs are practically the same but all topological minigaps are significantly larger. Other parameters as in Fig. 5.

Figure 11

Topological phase diagrams in the presence of generic mode mixing. (a) Topological phase diagram as a function of SO coupling $\alpha$ and Fermi energy $\tilde{\mu }$ for the tubular-core nanowire of Figs. 4 and 4 in the $n=1$ LP lobe (at $\mathrm{\Phi }=0.51{\mathrm{\Phi }}_0$). The color bar represents the number of MZMs $N_{\mathrm{M}}$. The boundary of the $m_J=0$ topological region without mode mixing is contoured with a blue line. (b) Same as (a) but for the $n=2$ LP lobe (at $\mathrm{\Phi }=1.51{\mathrm{\Phi }}_0$). [(c),(d)] Same as (a) and (b), respectively, but for the solid-core nanowire of Fig. 8. These phase diagrams are given as a function of radially averaged SO coupling $\langle \alpha \rangle$ and potential profile minimum $U_{\min }$, with $U_{\max }=0$ and $\mu =0\mathrm{meV}$.

Figure 12

Trivial and topological phases for the hollow-core model. Trivial (white) and topological (green) phases as a function of SO coupling $\alpha$ and Fermi energy $\tilde{\mu }$ for a hollow-core nanowire with $R=70$ nm, $d=0, {\mathrm{\Gamma }}_{\mathrm{S}}={\mathrm{\Delta }}_0$ and $g=0$. The green region shows the parameters for which $\tilde{\mu }>{\tilde{\mu }}_{\mathrm{c}}$, i.e., for which there can be topological phase transitions for some values of $\mathrm{\Phi }$. For $\alpha ={\alpha }_{\mathrm{c}}=-1/\left(2m^{*}R\right)$ the system is always trivial. The blue contour shows the topological phase diagram, i.e., the parameter regions for which there actually are Majorana ZEPs in the first lobe. Note that the upper blue contour is the same as Fig. 2. Other parameters as in Fig. 2.

Figure 13

Tubular-core model in the destructive Little-Parks regime. (a) Topological phase diagram as a function of constant $\alpha$ and $\mu$ for a tubular-core nanowire with $R=30\mathrm{nm}, d=0$, and $W=10\mathrm{nm}$. Parameter islands with topologically protected MZMs are bounded by red lines. The solid-orange curve has been calculated using the modified hollow-core approximation, with $R_{\mathrm{av}}=24.5\mathrm{nm}, \tilde{\mu }\in [-2.38,2.62]$, and ${\mathrm{\Gamma }}_{\mathrm{S}}^{\mathrm{av}}=1.9{\mathrm{\Delta }}_0$. (b) LDOS (in arbitrary units) as a function of $\omega$ and $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for the parameters of the red dot in (a): $\alpha =10\mathrm{meV}\mathrm{nm}$ and $\mu =39.1\mathrm{meV}$. ${\mathrm{\Gamma }}_{\mathrm{S}}=8{\mathrm{\Delta }}_0$ chosen so that the degeneracy points in the $n=0$ LP lobe remain around $\omega =\pm 0.18\mathrm{meV}$. There is a topological minigap $E_{\mathrm{g}}=20\mu \mathrm{eV}$ for $\mathrm{\Phi }=0.99{\mathrm{\Phi }}_0$. Other parameters as in Fig. 2.